This article relies largely or entirely on a
single source. (August 2021) |
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:
Thus
holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function.
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ln Γ(z) | ψ(0)(z) | ψ(1)(z) |
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ψ(2)(z) | ψ(3)(z) | ψ(4)(z) |
When m > 0 and Re z > 0, the polygamma function equals
where is the Hurwitz zeta function.
This expresses the polygamma function as the Laplace transform of (−1)m+1 tm/1 − e−t. It follows from Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)m+1 ψ(m)(x) is a completely monotone function.
Setting m = 0 in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the m = 0 case above but which has an extra term e−t/t.
It satisfies the recurrence relation
which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:
and
for all , where is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain uniquely to positive real numbers only due to their recurrence relation and one given function-value, say ψ(m)(1), except in the case m = 0 where the additional condition of strict monotonicity on is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionally. The case m = 0 must be treated differently because ψ(0) is not normalizable at infinity (the sum of the reciprocals doesn't converge).
where Pm is alternately an odd or even polynomial of degree |m − 1| with integer coefficients and leading coefficient (−1)m⌈2m − 1⌉. They obey the recursion equation
The multiplication theorem gives
and
for the digamma function.
The polygamma function has the series representation
which holds for integer values of m > 0 and any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as
This relation can for example be used to compute the special values [1]
Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.
One more series may be permitted for the polygamma functions. As given by Schlömilch,
This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:
Now, the natural logarithm of the gamma function is easily representable:
Finally, we arrive at a summation representation for the polygamma function:
Where δn0 is the Kronecker delta.
Also the Lerch transcendent
can be denoted in terms of polygamma function
The Taylor series at z = -1 is
and
which converges for |z| < 1. Here, ζ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.
These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments: [2]
and
where we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.
The hyperbolic cotangent satisfies the inequality
and this implies that the function
is non-negative for all m ≥ 1 and t ≥ 0. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that
is completely monotone. The convexity inequality et ≥ 1 + t implies that
is non-negative for all m ≥ 1 and t ≥ 0, so a similar Laplace transformation argument yields the complete monotonicity of
Therefore, for all m ≥ 1 and x > 0,
Since both bounds are strictly positive for , we have:
This can be seen in the first plot above.
For the case of the trigamma function () the final inequality formula above for , can be rewritten as:
so that for : .
This article relies largely or entirely on a
single source. (August 2021) |
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:
Thus
holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function.
![]() |
![]() |
![]() |
ln Γ(z) | ψ(0)(z) | ψ(1)(z) |
![]() |
![]() |
![]() |
ψ(2)(z) | ψ(3)(z) | ψ(4)(z) |
When m > 0 and Re z > 0, the polygamma function equals
where is the Hurwitz zeta function.
This expresses the polygamma function as the Laplace transform of (−1)m+1 tm/1 − e−t. It follows from Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)m+1 ψ(m)(x) is a completely monotone function.
Setting m = 0 in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the m = 0 case above but which has an extra term e−t/t.
It satisfies the recurrence relation
which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:
and
for all , where is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain uniquely to positive real numbers only due to their recurrence relation and one given function-value, say ψ(m)(1), except in the case m = 0 where the additional condition of strict monotonicity on is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionally. The case m = 0 must be treated differently because ψ(0) is not normalizable at infinity (the sum of the reciprocals doesn't converge).
where Pm is alternately an odd or even polynomial of degree |m − 1| with integer coefficients and leading coefficient (−1)m⌈2m − 1⌉. They obey the recursion equation
The multiplication theorem gives
and
for the digamma function.
The polygamma function has the series representation
which holds for integer values of m > 0 and any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as
This relation can for example be used to compute the special values [1]
Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.
One more series may be permitted for the polygamma functions. As given by Schlömilch,
This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:
Now, the natural logarithm of the gamma function is easily representable:
Finally, we arrive at a summation representation for the polygamma function:
Where δn0 is the Kronecker delta.
Also the Lerch transcendent
can be denoted in terms of polygamma function
The Taylor series at z = -1 is
and
which converges for |z| < 1. Here, ζ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.
These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments: [2]
and
where we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.
The hyperbolic cotangent satisfies the inequality
and this implies that the function
is non-negative for all m ≥ 1 and t ≥ 0. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that
is completely monotone. The convexity inequality et ≥ 1 + t implies that
is non-negative for all m ≥ 1 and t ≥ 0, so a similar Laplace transformation argument yields the complete monotonicity of
Therefore, for all m ≥ 1 and x > 0,
Since both bounds are strictly positive for , we have:
This can be seen in the first plot above.
For the case of the trigamma function () the final inequality formula above for , can be rewritten as:
so that for : .